nLab Sandbox

The Pin(2)-group is

$$S(\mathbb{C}) \cup \mathbf{j} S(\mathbb{C}) \,.$$

Identify the plane with the linear span of $\mathbf{j}$ and $\mathbf{k}$

$$\mathbb{R}^2 \;\simeq\; \langle \mathbf{j}, \mathbf{k}\rangle \;\subset\; \mathbb{H}_{\mathrm{im}} \,.$$

Then an element

$$R_{\alpha} \;\coloneqq\; Ad_{ \cos(\alpha/2) + \sin(\alpha/2) \mathbf{i} } \;\in\; S(\mathbb{C}) \subset S(\mathbb{H})$$

acts on $\mathbb{R}^2$ as

$$\begin{array}{rcl} R_\alpha(\mathbf{j}) &\equiv& \big( \cos(\alpha/2) + \sin(\alpha/2)\mathbf{i} \big) \, \mathbf{j} \, \big( \cos(\alpha/2) - \sin(\alpha/2)\mathbf{i} \big) \\ &=& \big( \cos(\alpha/2)^2 - \sin(\alpha/2)^2 \big) \, \mathbf{j} + \big( 2 \sin(\alpha/2) \cos(\alpha/2) \big) \, \mathbf{k} \\ &=& \cos(\alpha) \, \mathbf{j} + \sin(\alpha) \, \mathbf{k} \end{array}$$

(in the last line we used the sum-of-angles trigonometric identities)

while the element $I_y \coloneqq Ad_{\mathbf{j}}$ acts as

$$I_y \mathbf{j} \;=\; \mathbf{j} \, \mathbf{j} \, (- \mathbf{j}) \;=\; \mathbf{j}$$

$$I_y \mathbf{k} \;=\; \mathbf{j} \, \mathbf{k} \, (- \mathbf{j}) \;=\; - \mathbf{k}$$